Bundles over quantum weighted projective spaces

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1 Bundles over quantum weighted projective spaces Tomasz Swansea University Lancaster, September 2012 Joint work with Simon A Fairfax References: TB & SAF, Quantum teardrops, Comm. Math. Phys. in press (arxiv: ) TB & SAF, Bundles over quantum real weighted projective spaces, Axioms in press (arxiv: ) TB, Circle actions on a quantum Seifert manifold, Proceedings of Science (CORFU2011) (arxiv: )

2 Themes Examples of quantum orbifolds. Every n-orbifold is a quotient of a manifold by an almost free action of a subgroup of O(n). Every orbifold Riemannian surface is a quotient of a Seifert 3-manifold by an action of S 1. Illustration of noncommutative resolution of singularities.

3 Quantum spheres Let q be a real number, 0 < q < 1. The coordinate algebra O(Sq 2n+1 ) of the odd-dimensional quantum sphere is the unital complex -algebra with generators z 0, z 1,..., z n subject to the following relations: z i z j = qz j z i for i < j, z i z j = qz j z i for i j, z i z i = z i z i + (q 2 1) n m=i+1 z m z m, n z m zm = 1. m=0 The coordinate algebra O(S 2n q ) of the even-dimensional quantum sphere is the unital complex -algebra with generators z 0, z 1,..., z n and above relations supplemented with z n = z n.

4 For any n + 1 pairwise coprime numbers l 0,..., l n, O(U(1)) = C[u, u ] coacts on O(Sq 2n+1 ), as ϱ l0,...,l n : z i z i u l i, i = 0, 1,..., n. The quantum weighted projective space is the subalgebra of O(Sq 2n+1 ) containing all elements invariant under the coaction ϱ l0,...,l n, i.e. O(WP q (l 0, l 1,..., l n )) = {x O(Sq 2n+1 ) ϱ l0,...,l n (x) = x 1}. In the case l 0 = l 1 = = 1 one obtains the algebra of functions on the quantum complex projective space CP n q defined in [YaS Soibel man & LL Vaksman 90].

5 Quantum weighted projective lines The weighted projective line O(WP q (k, l)) is the -algebra generated by a and b subject to the following relations l 1 a = a, ab = q 2l ba, bb = q 2kl a k (1 q 2m a), Embedding in O(S 3 q): b b = a k l (1 q 2m a). m=1 m=0 a z 1 z 1, b zl 0 z k 1.

6 Quantum weighted projective lines.special cases O(WP q (1, 1)) is the standard Podleś sphere. O(WP q (1, l)) is the quantum teardrop. No repeated roots in polynomial relations. Classical singularities are resolved.

7 Representations Theorem Up to a unitary equivalence, the following is the list of all bounded irreducible -representations of O(WP q (k, l)). (i) One dimensional representation ϕ 0 : a 0, b 0. (ii) Infinite dimensional faithful representations ϕ s : O(WP q (k, l)) End(V s ), s = 1, 2,..., l. V s l 2 (N) has orthonormal basis e s p, p N, ϕ s (b)e s 0 = 0 and ϕ s (b)e s p = q k(lp+s) ϕ s (a)e s p = q 2(lp+s) e s p, l r=1 ( 1 q 2(lp+s r)) 1/2 e s p 1.

8 C(WP q (k, l)) and K -groups Theorem Let K s denote the algebra of all compact operators on the Hilbert space V s. There is a split-exact sequence of C -algebra maps 0 l s=1 K s C(WP q (k, l)) C 0. Consequently, K 0 (C(WP q (k, l))) = Z l+1, K 1 (C(WP q (k, l))) = 0.

9 Quantum principal bundles Definition Let H be a Hopf algebra with bijective antipode and let A be a right H-comodule algebra with coaction ϱ : A A H. Let B = A coh := {b A ϱ(b) = b 1}. A is a principal H-comodule algebra if: (a) the coaction ϱ is free, that is the canonical map can : A B A A H, a a aϱ(a ), is bijective (the Hopf-Galois condition); (b) there exists a strong connection in A, that is B A A, b a ba, splits as a left B-module and right H-comodule map (the equivariant projectivity).

10 Strong connection forms A right H-comodule algebra A with coaction ϱ : A A H is principal if and only if it admits a strong connection form, that is if there exists a map ω : H A A, such that ω(1) = 1 1, µ ω = η ε, (ω id) = (id ϱ) ω, (S ω) = (σ id) (ϱ id) ω. Here µ : A A A denotes the multiplication map, η : C A is the unit map, : H H H is the comultiplication, ε : H C counit and S : H H the (bijective) antipode of the Hopf algebra H, and σ : A H H A is the flip.

11 Freeness and almost freeness Theorem The algebra O(S 3 q) is a principal O(U(1))-comodule algebra over O(WP q (k, l)) by the coaction ϱ k,l if and only if k = l = 1 (the quantum Hopf fibration). Remark The coaction ϱ 1,l is almost free, meaning that the cokernel of the canonical map can is a finitely generated left O(S 3 q)-module.

12 Carving out the total space C[Z l ] coacts on O(S 3 q) by z 0 z 0 w, z 1 z 1 1. Coinvariants are the quantum lens space O(L q (l; 1, l)) [JH Hong & W Szymański 03]. It is generated by c = z l 0 and d = z 1, and admits an O(U(1))-coaction c c u, d d u. O(L q (l; 1, l)) is a non-cleft principal O(U(1))-comodule algebra over O(WP q (1, l)). Strong connection form: ω(u n )=c ω(u n 1 )c l ( ) l ( ) m q m(m+1) d m d m 1 ω(u n 1 )d. m q 2 m=1 (Non-cleft because multiples of 1 are the only invertible elements of O(S 3 q).)

13 Line bundles For any integer n, modules of sections on line bundles over O(WP q (1, l)) (associated to the principal comodule algebra O(L q (l; 1, l))) are defined as L[n] := {x O(L q (l; 1, l)) ϱ l (x) = x u n }. Each L[n] is a projective left O(WP q (1, l))-module with an idempotent E[n] given as follows. Write ω(u n ) = i ω(v n ) [1] i ω(u n ) [2] i. Then E[n] ij = ω(u n ) [2] iω(u n ) [1] j O(WP q (1, l)).

14 Index pairing Theorem For every s = 1, 2,..., l, (ϕ s, ϕ 0 ) is a 1-summable Fredholm module over O(WP q (k, l)) with the Chern character τ s (a m b n ) = for n = 0, m 0, and 0 otherwise. q2ms 1 q 2ml, Theorem For all s = 1, 2,..., l, τ s (Tr E[1]) = 1. Consequently, the left O(WP q (1, l))-module L[1] is not free.

15 Prolongations of quantum spheres Quantum spheres are right comodule algebras over the Hopf algebra O(Z 2 ) generated by a self-adjoint grouplike element v satisfying v 2 = 1. The coaction is z i z i v. O(U(1)) is a left O(Z 2 )-comodule algebra with coaction: u v u. Quantum spheres can be prolonged to algebras O(Σ m+1 q ) := O(Sq m ) O(Z2 )O(U(1)).

16 The structure of O(Σ m q ) Theorem (TB & BP Zieliński) 1 O(Σ 2n+2 q ) is an algebra isomorphic to O(Sq 2n+1 ) O(U(1)). 2 O(Σ 2n+1 q ) is isomorphic to a polynomial -algebra generated by ζ 0, ζ 1,..., ζ n and a central unitary ξ subject to the following relations ζ i ζ j = qζ j ζ i for i < j, ζ i ζ j = qζ j ζ i for i j, ζ i ζ i = ζ i ζ i + (q 2 1) 3 For all m > 1, Σ m+1 q bundles over RP m q. n m=i+1 ζ n = ζ n ξ. ζ m ζ m, n ζ m ζm = 1, m=0 are non-trivial quantum U(1)-principal

17 Weighted U(1)-actions on O(Σ 2n+1 q ) Choose n + 1 pairwise coprime numbers l 0,..., l n. O(Σ 2n+1 q ) is a right O(U(1))-comodule algebra with coaction Define ϱ l0,...,l n : ξ ξ u 2ln, ζ i ζ i u l i. O(RP q (l 0, l 1,..., l n )) := {x O(Σ 2n+1 q ) ϱ l0,...,l n (x) = x 1}.

18 Circle actions on the quantum Seifert manifold Σ 3 q The even case. O(RP q (2s, l)) is generated by a = ζ 2 1 ξ and c + = ζ l 0 ξs, which satisfy the following relations: a = a, ac + = q 2l c + a, c + c + = l 1 (1 q 2m a), c+c + = m=0 l (1 q 2m a). m=1 A -algebra generated by a, c + and above relations (with l odd) is denoted by O(RP 2 q(l; +)).

19 Circle actions on the quantum Seifert manifold Σ 3 q The odd case. O(RP q (2s 1, l)) is generated by a = ζ 2 1 ξ, b = ζl 0 ζ 1ξ s and c = ζ 2l 0 ξk, which satisfy the following relations: a = a, ab = q 2l ba, ac = q 4l c a, b 2 = q 3l ac, bc = q 2l c b, l 1 bb = q 2l a (1 q 2m a), b b = a b c = q l m=0 l (1 q 2m a), m=1 l l 1 (1 q 2m a)b, c b = q l b (1 q 2m a), m=1 2l 1 c c = (1 q 2m a), c c = m=0 2l m=1 m=0 (1 q 2m a). A -algebra generated by a, b, c and above relations is denoted by O(RP 2 q(l; )).

20 Observations RP q (1; +) is the quantum disc. RP q (1; ) is the quantum real projective plane RP 2 q [PM Hajac, R Matthes, W Szymański]. No multiple poles in polynomial relations for RP q (l; ±); the classical singularities are resolved.

21 Bounded representations of O(RP q (l; +)) There is a family of one-dimensional representations of O(RP 2 q(l; +)) labelled by θ [0, 1) and given by π θ (a) = 0, π θ (c + ) = e 2πiθ. All other representations are infinite dimensional, labelled by r = 1,..., l, and given by π r (a)e r n = q 2(ln+r) e r n, π r (c + )e r n = l m=1 ( 1 q 2(ln+r m)) 1/2 e r n 1, where e r n, n N is an orthonormal basis for the representation space H r = l 2 (N).

22 Bounded representations of O(RP q (l; )) A family of one-dimensional representations labelled by θ [0, 1): π θ (a) = 0, π θ (b) = 0, π θ (c ) = e 2πiθ. All other representations are infinite dimensional, labelled by r = 1,..., l: π r (a)e r n = q 2(ln+r) e r n, π r (c )e r n = π r (b)e r n = q ln+r l m=1 2l m=1 ( 1 q 2(ln+r m)) 1/2 e r n 2, ( 1 q 2(ln+r m)) 1/2 e r n 1, where e r n, n N, is an orthonormal basis for the representation space H r = l 2 (N).

23 K -theory of C(RP q (l; ±)) Let J ± be ideals of C(RP q (l; ±)) generated by a; p ± : C(RP q (l; ±)) C(RP q (l; ±))/J ±. C(RP q (l; ±))/J ± is generated by p ± (c ± ) and C(RP q (l; ±))/J ± = C(S 1 ). There are short exact sequences: 0 l r=1 K r i ± C(RP q (l; ±)) p ± C(S 1 ) 0. The six-term sequences of the K -groups lead to 0 K 1 (C(RP q (l; ±))) Z δ ± Z l K 0 (i ± ) K 0 (C(RP q (l; ±))) K 0(p ± ) Z 0.

24 K -theory of C(RP q (l; ±)) The connecting map δ + comes out as δ + : Z Z l, m (m, m,..., m). Since δ + is injective, K 1 (C(RP q (l; +))) = 0. Furthermore, K 0 (C(RP q (l; +))) = Im(K 0 (i + )) Z = coker(δ + ) Z = Z l, The connecting map δ comes out as δ : Z Z l, m (2m, 2m,..., 2m). Since δ is injective, K 1 (C(RP q (l; ))) = 0. Furthermore, K 0 (C(RP q (l; ))) = coker(δ ) Z = Z 2 Z l,

25 The structure of C(RP q (l; +)) Starting from 0 K T σ C(S 1 ) 0, one finds C(RP q (l; +)) = T σ T σ σ T }{{} l times = C(D q ) σ C(D q ) σ σ C(D q ). }{{} l times The symbol map σ can be understood as a projection induced by the inclusion of the classical circle as the boundary of the quantum disc. C(RP q (l; +)) is a quantum double suspension of l points [Hong-Szymański].

26 The structure of C(RP q (l; )) Starting from 0 K C(RP 2 q) σ C(S 1 ) 0 [Hajac-Matthes-Szymański], one finds C(RP q (l; )) = C(RP 2 q) σ C(RP 2 q) σ σ C(RP 2 q), }{{} l times where the map σ is induced from the projection that corresponds to the inclusion of the classical circle as the equator of the quantum sphere S 2 q.

27 Freeness and almost freeness Theorem The algebra O(Σ 3 q) is a principal O(U(1))-comodule algebra over O(RP q (k, l)) by the coaction ϱ k,l if and only if k = l = 1 (the circle bundle over RP 2 q). Remark For O(RP q (l; ±)), the cokernel of the canonical map can : O(Σ 3 q) O(RPq(l;±)) O(Σ 3 q) O(Σ 3 q) O(U(1)), (corresponding to ϱ k,l, k = 1, 2) is a finitely generated left O(Σ 3 q)-module.

28 Carving out total spaces C[Z l ] coacts on O(Σ 3 q) by ζ 0 ζ 0 w, ζ 1 ζ 1 1, ξ ξ 1. Algebra of coinvariants O(Σ 3 q,l ) is generated by α = ζl 0 and ζ 1, ξ, and admits an O(U(1))-coaction α α u, ζ 1 ζ 1 u, ξ ξ u 2. O(Σ 3 q,l ) is a non-cleft principal O(U(1))-comodule algebra over O(RP q (l; )). Strong connection form: ω(u n )=α ω(u n 1 )α l ( ) l ( ) m q m(m+1) ζ 2m 1 m 1 ξ m ω(u n 1 )ζ 1 q 2 m=1

29 Carving out total spaces C[Z 2l ] (l-odd) coacts on O(Σ 3 q) by ζ 0 ζ 0 w 2, ζ 1 ζ 1 w l, ξ ξ 1. Algebra of coinvariants O(Σ 3 q,l+ ) is generated by α = ζl 0, β = ζ1 2 and ξ and admits an O(U(1))-coaction α α u, β β u, ξ ξ u. O(Σ 3 q,l+ ) is a cleft (trivial) principal O(U(1))-comodule algebra over O(RP q (l; )). The cleaving map is: O(U(1)) O(Σ 3 q,l+ ), u ξ.

30 In progress... Weighted circle actions on the Heegaard-type quantum sphere. Interpretation of weighted projective spaces in terms of graph C -algebras. Development of differential structures on weighted projective spaces. Spectral geometry of quantum weighted projective spaces.

(Not only) Line bundles over noncommutative spaces

(Not only) Line bundles over noncommutative spaces Giovanni Landi Trieste Gauge Theory and Noncommutative Geometry Radboud University Nijmegen ; April 4 8, 2016 Work done over few years with Francesca